Theorem 1 Let be a compact connected group, with a Haar probability measure . Let be compact subsets of . Then

Remark 1 The estimate is sharp, as can be seen by considering the case when is a unit circle, and are arcs; similarly if is any compact connected group that projects onto the circle. The connectedness hypothesis is essential, as can be seen by considering what happens if and are a non-trivial open subgroup of . For locally compact connected groups which are unimodular but not compact, there is an analogous statement, but with now a Haar measure instead of a Haar probability measure, and the right-hand side replaced simply by . The case when is a torus is due to Macbeath, and the case when is a circle is due to Raikov. The theorem is closely related to the Cauchy-Davenport inequality; indeed, it is not difficult to use that inequality to establish the circle case, and the circle case can be used to deduce the torus case by considering increasingly dense circle subgroups of the torus (alternatively, one can also use Kneser’s theorem).

By inner regularity, the hypothesis that are compact can be replaced with Borel measurability, so long as one then adds the additional hypothesis that is also Borel measurable.

A short proof of Kemperman’s theorem was given by Ruzsa. In this post I wanted to record how this argument can be used to establish the following more “robust” version of Kemperman’s theorem, which not only lower bounds , but gives many elements of some multiplicity:

Theorem 2 Let be a compact connected group, with a Haar probability measure . Let be compact subsets of . Then for any , one has

Indeed, Theorem 1 can be deduced from Theorem 2 by dividing (1) by and then taking limits as . The bound in (1) is sharp, as can again be seen by considering the case when are arcs in a circle. The analogous claim for cyclic groups for prime order was established by Pollard, and for general abelian groups by Green and Ruzsa.

Let us now prove Theorem 2. It uses a submodularity argument related to one discussed in this previous post. We fix and with , and define the quantity

for any compact set . Our task is to establish that whenever .

We first verify the extreme cases. If , then , and so in this case (since ). At the other extreme, if , then from the inclusion-exclusion principle we see that , and so again in this case.

To handle the intermediate regime when lies between and , we rely on the submodularity inequality

for arbitrary compact . This inequality comes from the obvious pointwise identity

whence

and thus (noting that the quantities on the left are closer to each other than the quantities on the right)

at which point (2) follows by integrating over and then using the inclusion-exclusion principle.

Now introduce the function

for . From the preceding discussion vanishes at the endpoints ; our task is to show that is non-negative in the interior region . Suppose for contradiction that this was not the case. It is easy to see that is continuous (indeed, it is even Lipschitz continuous), so there must be at which is a local minimum and not locally constant. In particular, . But for any with , we have the translation-invariance

Note that depends continuously on , equals when is the identity, and has an average value of . As is connected, we thus see from the intermediate value theorem that for any , we can find such that , and thus by inclusion-exclusion . By definition of , we thus have

Taking infima in (and noting that the hypotheses on are independent of ) we conclude that

for all . As is a local minimum and is arbitrarily small, this implies that is locally constant, a contradiction. This establishes Theorem 2.

We observe the following corollary:

Corollary 3 Let be a compact connected group, with a Haar probability measure . Let be compact subsets of , and let . Then one has the pointwise estimate

if , and

if .

Once again, the bounds are completely sharp, as can be seen by computing when are arcs of a circle. For quasirandom , one can do much better than these bounds, as discussed in this recent blog post; thus, the abelian case is morally the worst case here, although it seems difficult to convert this intuition into a rigorous reduction.

Proof: By cyclic permutation we may take . For any

we can bound

where we used Theorem 2 to obtain the third line. Optimising in , we obtain the claim.

Theorem 1 (Noncommutative Freiman-Kneser theorem for small doubling) Let , and let be a finite non-empty subset of a multiplicative group such that for some finite set of cardinality at least , where is the product set of and . Then there exists a finite subgroup of with cardinality , such that is covered by at most right-cosets of , where depend only on .

One can of course specialise here to the case , and view this theorem as a classification of those sets of doubling constant at most .

In fact Hamidoune’s argument, which is completely elementary, gives the very nice explicit constants and , which are essentially optimal except for factors of (as can be seen by considering an arithmetic progression in an additive group). This result was also independently established (in the case) by Tom Sanders (unpublished) by a more Fourier-analytic method, in particular drawing on Sanders’ deep results on the Wiener algebra on arbitrary non-commutative groups .

This type of result had previously been known when was less than the golden ratio , as first observed by Freiman; see my previous blog post for more discussion.

Theorem 1 is not, strictly speaking, contained in Hamidoune’s paper, but can be extracted from his arguments, which share some similarity with the recent simple proof of the Ruzsa-Plünnecke inequality by Petridis (as discussed by Tim Gowers here), and this is what I would like to do below the fold. I also include (with permission) Sanders’ unpublished argument, which proceeds instead by Fourier-analytic methods. Read the rest of this entry »

In Notes 3, we saw that the number of additive patterns in a given set was (in principle, at least) controlled by the Gowers uniformity norms of functions associated to that set.

Such norms can be defined on any finite additive group (and also on some other types of domains, though we will not discuss this point here). In particular, they can be defined on the finite-dimensional vector spaces over a finite field .

In this case, the Gowers norms are closely tied to the space of polynomials of degree at most . Indeed, as noted in Exercise 20 of Notes 4, a function of norm has norm equal to if and only if for some ; thus polynomials solve the “ inverse problem” for the trivial inequality . They are also a crucial component of the solution to the “ inverse problem” and “ inverse problem”. For the former, we will soon show:

Proposition 1 ( inverse theorem for ) Let be such that and for some . Then there exists such that , where is a constant depending only on .

Thus, for the Gowers norm to be almost completely saturated, one must be very close to a polynomial. The converse assertion is easily established:

Exercise 1 (Converse to inverse theorem for ) If and for some , then , where is a constant depending only on .

In the world, one no longer expects to be close to a polynomial. Instead, one expects to correlate with a polynomial. Indeed, one has

Lemma 2 (Converse to the inverse theorem for ) If and are such that , where , then .

Proof: From the definition of the norm (equation (18) from Notes 3), the monotonicity of the Gowers norms (Exercise 19 of Notes 3), and the polynomial phase modulation invariance of the Gowers norms (Exercise 21 of Notes 3), one has

and the claim follows.

In the high characteristic case at least, this can be reversed:

Theorem 3 ( inverse theorem for ) Suppose that . If is such that and , then there exists such that .

This result is sometimes referred to as the inverse conjecture for the Gowers norm (in high, but bounded, characteristic). For small , the claim is easy:

Exercise 2 Verify the cases of this theorem. (Hint: to verify the case, use the Fourier-analytic identities and , where is the space of all homomorphisms from to , and are the Fourier coefficients of .)

This conjecture for larger values of are more difficult to establish. The case of the theorem was established by Ben Green and myself in the high characteristic case ; the low characteristic case was independently and simultaneously established by Samorodnitsky. The cases in the high characteristic case was established in two stages, firstly using a modification of the Furstenberg correspondence principle, due to Ziegler and myself. to convert the problem to an ergodic theory counterpart, and then using a modification of the methods of Host-Kra and Ziegler to solve that counterpart, as done in this paper of Bergelson, Ziegler, and myself.

The situation with the low characteristic case in general is still unclear. In the high characteristic case, we saw from Notes 4 that one could replace the space of non-classical polynomials in the above conjecture with the essentially equivalent space of classical polynomials . However, as we shall see below, this turns out not to be the case in certain low characteristic cases (a fact first observed by Lovett, Meshulam, and Samorodnitsky, and independently by Ben Green and myself), for instance if and ; this is ultimately due to the existence in those cases of non-classical polynomials which exhibit no significant correlation with classical polynomials of equal or lesser degree. This distinction between classical and non-classical polynomials appears to be a rather non-trivial obstruction to understanding the low characteristic setting; it may be necessary to obtain a more complete theory of non-classical polynomials in order to fully settle this issue.

The inverse conjecture has a number of consequences. For instance, it can be used to establish the analogue of Szemerédi’s theorem in this setting:

Theorem 4 (Szemerédi’s theorem for finite fields) Let be a finite field, let , and let be such that . If is sufficiently large depending on , then contains an (affine) line for some with .

Exercise 3 Use Theorem 4 to establish the following generalisation: with the notation as above, if and is sufficiently large depending on , then contains an affine -dimensional subspace.

We will prove this theorem in two different ways, one using a density increment method, and the other using an energy increment method. We discuss some other applications below the fold.

Theorem 1 (Plünnecke-Ruzsa inequality) Let be finite non-empty subsets of an additive group , such that for all and some scalars . Then there exists a subset of such that .

The proof uses graph-theoretic techniques. Setting , we obtain a useful corollary: if has small doubling in the sense that , then we have for all , where is the sum of copies of .

In a recent paper, I adapted a number of sum set estimates to the entropy setting, in which finite sets such as in are replaced with discrete random variables taking values in , and (the logarithm of) cardinality of a set is replaced by Shannon entropy of a random variable . (Throughout this note I assume all entropies to be finite.) However, at the time, I was unable to find an entropy analogue of the Plünnecke-Ruzsa inequality, because I did not know how to adapt the graph theory argument to the entropy setting.

Theorem 2 (Entropy Plünnecke-Ruzsa inequality) Let be independent random variables of finite entropy taking values in an additive group , such that for all and some scalars . Then .

In fact Theorem 2 is a bit “better” than Theorem 1 in the sense that Theorem 1 needed to refine the original set to a subset , but no such refinement is needed in Theorem 2. One corollary of Theorem 2 is that if , then for all , where are independent copies of ; this improves slightly over the analogous combinatorial inequality. Indeed, the function is concave (this can be seen by using the version of Theorem 2 (or (2) below) to show that the quantity is decreasing in ).

Theorem 2 is actually a quick consequence of the submodularity inequality

in information theory, which is valid whenever are discrete random variables such that and each determine (i.e. is a function of , and also a function of ), and and jointly determine (i.e is a function of and ). To apply this, let be independent discrete random variables taking values in . Observe that the pairs and each determine , and jointly determine . Applying (1) we conclude that

which after using the independence of simplifies to the sumset submodularity inequality

(this inequality was also recently observed by Madiman; it is the case of Theorem 2). As a corollary of this inequality, we see that if , then

The proof of Theorem 2 seems to be genuinely different from the graph-theoretic proof of Theorem 1. It would be interesting to see if the above argument can be somehow adapted to give a stronger version of Theorem 1. Note also that both Theorem 1 and Theorem 2 have extensions to more general combinations of than ; see this paper and this paper respectively.

Motivation: In the more quantitative areas of mathematics, such as analysis and combinatorics, one has to frequently keep track of a large number of exponents in one’s identities, inequalities, and estimates. For instance, if one is studying a set of N elements, then many expressions that one is faced with will often involve some power of N; if one is instead studying a function f on a measure space X, then perhaps it is an norm which will appear instead. The exponent involved will typically evolve slowly over the course of the argument, as various algebraic or analytic manipulations are applied. In some cases, the exact value of this exponent is immaterial, but at other times it is crucial to have the correct value of at hand. One can (and should) of course carefully go through one’s arguments line by line to work out the exponents correctly, but it is all too easy to make a sign error or other mis-step at one of the lines, causing all the exponents on subsequent lines to be incorrect. However, one can guard against this (and avoid some tedious line-by-line exponent checking) by continually calibrating these exponents at key junctures of the arguments by using basic examples of the object of study (sets, functions, graphs, etc.) as test cases. This is a simple trick, but it lets one avoid many unforced errors with exponents, and also lets one compute more rapidly.

Quick description: When trying to quickly work out what an exponent p in an estimate, identity, or inequality should be without deriving that statement line-by-line, test that statement with a simple example which has non-trivial behaviour with respect to that exponent p, but trivial behaviour with respect to as many other components of that statement as one is able to manage. The “non-trivial” behaviour should be parametrised by some very large or very small parameter. By matching the dependence on this parameter on both sides of the estimate, identity, or inequality, one should recover p (or at least a good prediction as to what p should be).

General discussion: The test examples should be as basic as possible; ideally they should have trivial behaviour in all aspects except for one feature that relates to the exponent p that one is trying to calibrate, thus being only “barely” non-trivial. When the object of study is a function, then (appropriately rescaled, or otherwise modified) bump functions are very typical test objects, as are Dirac masses, constant functions, Gaussians, or other functions that are simple and easy to compute with. In additive combinatorics, when the object of study is a subset of a group, then subgroups, arithmetic progressions, or random sets are typical test objects. In graph theory, typical examples of test objects include complete graphs, complete bipartite graphs, and random graphs. And so forth.

This trick is closely related to that of using dimensional analysis to recover exponents; indeed, one can view dimensional analysis as the special case of exponent calibration when using test objects which are non-trivial in one dimensional aspect (e.g. they exist at a single very large or very small length scale) but are otherwise of a trivial or “featureless” nature. But the calibration trick is more general, as it can involve parameters (such as probabilities, angles, or eccentricities) which are not commonly associated with the physical concept of a dimension. And personally, I find example-based calibration to be a much more satisfying (and convincing) explanation of an exponent than a calibration arising from formal dimensional analysis.

When one is trying to calibrate an inequality or estimate, one should try to pick a basic example which one expects to saturate that inequality or estimate, i.e. an example for which the inequality is close to being an equality. Otherwise, one would only expect to obtain some partial information on the desired exponent p (e.g. a lower bound or an upper bound only). Knowing the examples that saturate an estimate that one is trying to prove is also useful for several other reasons – for instance, it strongly suggests that any technique which is not efficient when applied to the saturating example, is unlikely to be strong enough to prove the estimate in general, thus eliminating fruitless approaches to a problem and (hopefully) refocusing one’s attention on those strategies which actually have a chance of working.

Calibration is best used for the type of quick-and-dirty calculations one uses when trying to rapidly map out an argument that one has roughly worked out already, but without precise details; in particular, I find it particularly useful when writing up a rapid prototype. When the time comes to write out the paper in full detail, then of course one should instead carefully work things out line by line, but if all goes well, the exponents obtained in that process should match up with the preliminary guesses for those exponents obtained by calibration, which adds confidence that there are no exponent errors have been committed.

Additive combinatorics is largely focused on the additive properties of finite subsets A of an additive group . This group can be finite or infinite, but there is a very convenient trick, the Ruzsa projection trick, which allows one to reduce the latter case to the former. For instance, consider the set inside the integers . The integers of course form an infinite group, but if we are only interested in sums of at most two elements of A at a time, we can embed A ininside the finite cyclic group without losing any combinatorial information. More precisely, there is a Freiman isomorphism of order 2 between the set in and the set in . One can view the latter version of as a model for the former version of . More generally, it turns out that any finite set A in an additive group can be modeled in the above set by an equivalent set in a finite group, and in fact one can ensure that this ambient modeling group is not much larger than A itself if A has some additive structure; see this paper of Ruzsa (or Lemma 5.26 of my book with Van Vu) for a precise statement. This projection trick has a number of important uses in additive combinatorics, most notably in Ruzsa’s simplified proof of Freiman’s theorem.

Given the interest in non-commutative analogues of Freiman’s theorem, it is natural to ask whether one can similarly model finite sets A in multiplicative (and non-commutative) groups using finite models. Unfortunately (as I learned recently from Akshay Venkatesh, via Ben Green), this turns out to be impossible in general, due to an old example of Higman. More precisely, Higman shows:

Theorem 1. There exists an infinite group G generated by four distinct elements a,b,c,d that obey the relations

; (1)

in fact, a and c generate the free group in G. On the other hand, if G’ is a finite group containing four elements a,b,c,d obeying (1), then a,b,c,d are all trivial.

As a consequence, the finite set in G has no model (in the sense of Freiman isomorphisms) in a finite group.

Theorem 1 is proven by a small amount of elementary group theory and number theory, and it was neat enough that I thought I would reproduce it here.

I’ve uploaded a new paper to the arXiv entitled “The sum-product phenomenon in arbitrary rings“, and submitted to Contributions to Discrete Mathematics. The sum-product phenomenon asserts, very roughly speaking, that given a finite non-empty set A in a ring R, then either the sum set or the product set will be significantly larger than A, unless A is somehow very close to being a subring of R, or if A is highly degenerate (for instance, containing a lot of zero divisors). For instance, in the case of the integers , which has no non-trivial finite subrings, a long-standing conjecture of Erdös and Szemerédi asserts that for every finite non-empty and every . (The current best result on this problem is a very recent result of Solymosi, who shows that the conjecture holds for any greater than 2/3.) In recent years, many other special rings have been studied intensively, most notably finite fields and cyclic groups, but also the complex numbers, quaternions, and other division algebras, and continuous counterparts in which A is now (for instance) a collection of intervals on the real line. I will not try to summarise all the work on sum-product estimates and their applications (which range from number theory to graph theory to ergodic theory to computer science) here, but I discuss this in the introduction to my paper, which has over 50 references to this literature (and I am probably still missing out on a few).

I was recently asked the question as to what could be said about the sum-product phenomenon in an arbitrary ring R, which need not be commutative or contain a multiplicative identity. Once one makes some assumptions to avoid the degenerate case when A (or related sets, such as A-A) are full of zero-divisors, it turns out that there is in fact quite a bit one can say, using only elementary methods from additive combinatorics (in particular, the Plünnecke-Ruzsa sum set theory). Roughly speaking, the main results of the paper assert that in an arbitrary ring, a set A which is non-degenerate and has small sum set and product set must be mostly contained inside a subring of R of comparable size to A, or a dilate of such a subring, though in the absence of invertible elements one sometimes has to enlarge the ambient ring R slightly before one can find the subring. At the end of the paper I specialise these results to specific rings, such as division algebras or products of division algebras, cyclic groups, or finite-dimensional algebras over fields. Generally speaking, the methods here give very good results when the set of zero divisors is sparse and easily describable, but poor results otherwise. (In particular, the sum-product theory in cyclic groups, as worked out by Bourgain and coauthors, is only recovered for groups which are the product of a bounded number of primes; the case of cyclic groups whose order has many factors seems to require a more multi-scale analysis which I did not attempt to perform in this paper.)Read the rest of this entry »

This week I am visiting the University of Washington in Seattle, giving the Milliman Lecture Series for 2007-2008. My chosen theme here is “Recent developments in arithmetic combinatorics“. In my first lecture, I will speak (once again) on how methods in additive combinatorics have allowed us to detect additive patterns in the prime numbers, in particular discussing my joint work with Ben Green. In the second lecture I will discuss how additive combinatorics has made it possible to study the invertibility and spectral behaviour of random discrete matrices, in particular discussing my joint work with Van Vu; and in the third lecture I will discuss how sum-product estimates have recently led to progress in the theory of expanders relating to Lie groups, as well as to sieving over orbits of such groups, in particular presenting work of Jean Bourgain and his coauthors.

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